This paper proposes a systematic formalization of Irifuji Motoyoshi's circular model of actu-re-ality—developed in The Problem of Actu-Re-ality (2020) and The Ultima Thule of Actu-Re-ality (2025) —using the language of homological algebra, category theory, derived functors, and spectral sequences. The circular model describes how the modalities of reality (manifest actu-re-ality, possibility, latency, and the liberation of negation) are connected through a cyclic but irreversible structure. We demonstrate that this structure admits a natural formulation as a chain complex of modules over a non-commutative "reality ring, " whose homology groups precisely capture the philosophical core concepts of asymmetry, residue, and emergence. Key Highlights: • Classification of Emergence: Weak, strong, and super-strong emergence are strictly classified via derived functors (Ext and Tor) and infinite projective dimension. • Universality: The model's claim to function as a universal "game board" is formalized through adjoint functors and monads. • The Starting-Point Problem: The infinite regress of the origin is resolved via projective limits and the non-vanishing of ¹ (failure of the Mittag-Leffler condition). • Interdisciplinary Connections: The paper establishes deep structural isomorphisms with linear logic (resource semantics), spontaneous symmetry breaking (SSB) in physics, and global obstructions in sheaf cohomology.
Takayuki Shoji (Fri,) studied this question.